Abstract. In a recent paper by P.W.Jones and J.Journ\'e ({\it On weak
convergence in $H^{1}(R^{d})$ }, Proc. Amer. Math. Soc. 120
(1994) , 137 - 138. ) , there are some inaccuracies. In this
note, we give a counterexample to his theorem and a
corrected form of the limit's theorem.